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Published in:
2015 | OriginalPaper | Chapter
7. Harmonic Maps
Author : Sören Bartels
Published in: Numerical Methods for Nonlinear Partial Differential Equations
Publisher: Springer International Publishing
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Abstract
Partial differential equations with a nonlinear pointwise equality constraint arise in the mathematical modeling of liquid crystals and ferromagnets. The solutions are vector fields that describe the orientation of the molecules or the magnetization field. A simple model problem defines harmonic maps in the sphere as minimizers of the Dirichlet energy subject to a pointwise unit-length constraint. Lowest-order conforming finite element methods prohibit imposing the constraint everywhere and limited regularity and uniqueness properties of harmonic maps require a careful numerical treatment. The stability and convergence of iterative methods that employ linearizations and projections are proved, and short implementations are provided in this chapter. Extensions to approximating related evolution problems, such as the harmonic map heat flow and wave maps are discussed.